Electric Field of Uniformly Charged Ring: Derivation Explained

[dido28]

hi every one on an exercise on book we ask us to find the electrical field for a uniformly charged ring where we going to find : E=kqz/(z²+R²)^3/2
then we have to derivate it wth respect to z and we find : dE/dz=kq*(R²+2z²)/(z²+R²)^5/2
so my question is how do we get this derivation ?

 

[WannabeNewton]

You will have show us what you have *tried* thus far. Where did you get stuck in the derivation when you tried it yourself? Also note that the given expression is only the electric field produced by the uniformly charged static ring at points along its symmetry axis (in case you didn't know already).

 

[dido28]

well me i found : dE/dz=qk[ (z²+R²)^3/2-3z²(z²+R²)^-1/2 ]/(z²+R²)³
i just applied the rule that i know but in the book it seems that they did some simplification so my question is how they did this

 

[WannabeNewton]

Oh are you only interested in finding the derivative? Do you already know how to get the electric field itself?

 

[dido28]

yes that's it

 

[WannabeNewton]

Are you sure it wasn't ##R^2 - 2z^2## as opposed to ##R^2 + 2z^2## in the numerator of the final answer?

 

[dido28]

oh yes sorry it's R²-2z² :shy:

 

[WannabeNewton]

Ok. Would you agree that, before doing any simplifications, the derivative comes out to ##\frac{\mathrm{d} E}{\mathrm{d} z} = \frac{kq}{(z^2 + R^2)^{3/2}} - \frac{3kqz^2}{(z^2 + R^2)^{5/2}}##? All I have done is use the product and chain rule; I haven't done any simplifications at all. Now, can you find a way to combine these two expressions?

 

[dido28]

yes it's done i get the result . thanks for your help WannabeNewton

 

[WannabeNewton]

I didn't do anything mate! It was all you :) Good luck with your studies.